https://file.aiquickdraw.com/mathbot/file/1730931121807wiu0x116.jpg

To determine the values of \( A \) and \( B \) in the transformation \( g(x) = A f(Bx) \), let’s analyze the changes between the graphs of \( f(x) \) and \( g(x) \).

### Step 1: Analyzing \( f(x) \)
In the first graph, \( f(x) \) has a V-shape, which typically suggests an absolute value function, probably something like \( f(x) = |x| \) or a line with a sharp point at the origin.

### Step 2: Observing \( g(x) \)
In the second graph, \( g(x) \), we notice that:
1. The V-shape is preserved, indicating that the function structure remains similar.
2. The graph of \( g(x) \) is stretched vertically and appears to have a different slope on each side of the V compared to \( f(x) \).

### Step 3: Determining \( A \) (Vertical Scaling)
Looking at the vertical scaling, we can see that the graph of \( g(x) \) is stretched. The slope of each side of the V in \( f(x) \) appears to double in \( g(x) \), which suggests a vertical stretch factor of 2. Thus, \( A = 2 \).

### Step 4: Determining \( B \) (Horizontal Scaling)
Since the point where the V-shape changes direction is not shifted horizontally, we conclude that there is no horizontal compression or stretch. Therefore, \( B = 1 \).

### Conclusion
The values are:
- \( A = 2 \)
- \( B = 1 \)

Would you like a deeper breakdown of each step or have any specific questions?

Here are some follow-up questions you might explore:
1. What other types of transformations can apply to \( f(x) \)?
2. How does changing \( B \) affect the graph of \( f(x) \)?
3. What effect does a negative value for \( A \) have on the graph?
4. Can you identify transformations that involve both a horizontal and vertical shift?
5. How would you determine the transformation if the graph were shifted horizontally?

**Tip:** Remember that vertical transformations (affecting \( A \)) change the "height" of the graph, while horizontal transformations (affecting \( B \)) change the "width" or compression/stretching along the x-axis.

The graph above shows the function \( f(x) \). The graph below shows \( g(x) \).

\( g(x) \) is a transformation of \( f(x) \) where \( g(x) = A f(Bx) \). Find the values of \( A \) and \( B \).

Understand how to find the transformation constants A and B for the function \( g(x) = A f(Bx) \), given the graph of \( f(x) \) and its transformation \( g(x) \). Learn about vertical and horizontal stretching with step-by-step explanations.

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